📚 Year 11 CIE Further Maths: Vocabulary Terminology Quick Memorisation Guide | 十一年级CIE进阶数学：词汇术语速记指南
Mastering the vocabulary of CIE Further Mathematics (IGCSE 0606) is half the battle. Once you can read a question and instantly decode terms like 'discriminant', 'asymptote', 'binomial expansion' or 'definite integral', the underlying calculations become far less intimidating. This guide pairs high-frequency technical terms with simple memory hooks, grouping them by topic to help your brain build meaningful connections rather than just memorise lists.
掌握CIE进阶数学（IGCSE 0606）的词汇是成功的一半。一旦你能读懂题目并立刻破解诸如'discriminant'（判别式）、'asymptote'（渐近线）、'binomial expansion'（二项式展开）或'definite integral'（定积分）这样的术语，背后的计算就会变得不那么可怕。本指南将高频专业术语与简单的记忆钩子配对，并按主题分组，帮助你的大脑建立有意义的联系，而不是仅仅背诵清单。
1. Functions and Relations | 函数与关系
Domain – think 'do' input: the set of all allowable x-values you are allowed to 'do' something with. Range – the resulting y-values after the function 'runs'. A one-to-one function maps each x to a unique y and passes the horizontal line test; remember it as 'one partner each'. A composite function written as fg(x) means 'first apply g, then f' – the order matters, like putting on socks before shoes. The inverse function f⁻¹(x) undoes the original: if f(x) = 2x + 3, then f⁻¹ undoes the '+3' and '×2' by reversing them. The notation f⁻¹ does not mean 1/f. Modulus function |x| always gives a non-negative output; its graph is a V shape and any equation with |...| may need you to split into positive and negative cases.
定义域（Domain）—— 联想'do'（可做的输入）：函数允许输入的所有x值的集合。 值域（Range）—— 函数'运行'后产生的y值。 一一函数（one-to-one function）将每个x映射到唯一的y，并能通过水平线检验；可记为'一人一个搭档'。 复合函数（composite function）写作fg(x)，表示'先作用g，再作用f'——顺序很重要，就像先穿袜子再穿鞋。 反函数（inverse function）f⁻¹(x)会撤销原函数：若f(x) = 2x + 3，则f⁻¹通过逆向操作撤销'+3'和'×2'。符号f⁻¹不是1/f的意思。 绝对值函数（modulus function）|x|总是给出非负输出；其图像是V形，任何带|...|的方程都可能需要拆分成正负两种情况。
2. Quadratics and Discriminants | 二次函数与判别式
Quadratic comes from Latin 'quadratus' meaning square, so any ax² + bx + c. The discriminant Δ = b² − 4ac 'discriminates' ts, Δ = 0 → one repeated real root, Δ < 0 → no real roots (two complex roots). A mnemonic: 'Positive – Pair, Zero – Same, Negative – None' for real roots. Completing the square rewrites a quadratic as a(x + p)² + q, which reveals the vertex (minimum or maximum point). The quadratic formula x = [−b ± √b² − 4ac] / 2a gives the solutions directly. For inequalities, sketching the parabola helps: the solution to ax² + bx + c > 0 lies outside the roots if a > 0.
二次函数（Quadratic）源自拉丁语'quadratus'（正方形），因此指任何形如ax² + bx + c的式子。 判别式（discriminant）Δ = b² − 4ac，它'辨别'根的类型：若Δ > 0 → 两个不同的实根，Δ = 0 → 一个重根，Δ < 0 → 无实根（两个复根）。口诀：'正数——成对，零——相同，负数——没有'（针对实根）。 配方法（Completing the square）将二次式改写为a(x + p)² + q，从而揭示顶点（vertex，即最小值或最大值点）。 求根公式（quadratic formula）x = [−b ± √b² − 4ac] / 2a 直接给出解。处理不等式（inequalities）时，画抛物线草图会有帮助：若a > 0，ax² + bx + c > 0的解在两根之外。
3. Exponentials and Logarithms | 指数与对数
Exponential functions have the form y = aˣ where the variable is in the exponent; they grow or decay very fast. The natural exponential function y = eˣ uses Euler's number e ≈ 2.718. Logarithms are the inverse of exponentials: if bˣ = y then log_b(y) = x. The natural logarithm y log laws: log(ab) = log a + log b, log(a/b) = log a − log b, log aⁿ = n log a. Remember: 'logs turn multiplication into addition' – g x) and e (natural log, written ln x). The change-of-base formula log_a b = (log_c b)/(log_c a) lets you use calculator-friendly bases.
指数函数（Exponential functions）的形式为y = aˣ，变量位于指数位置；它们增长或衰减得非常快。 自然指数函数（natural exponential function）y = eˣ 使用欧拉数e ≈ 2.718。 对数（Logarithms）是指数的逆运算：若bˣ = y，则log_b(y) = x。 自然对数（natural logarithm）ln x 表示log_e x。关键对数律：log(ab) = log a + log b，log(a/b) = log a − log b，log aⁿ = n log a。记住：'对数化乘为加'——这能极大节省时间。求解指数方程时，两边取ln并运用幂法则。两个特殊底数是10（常用对数，写作log x）和e（自然对数，写作ln x）。换底公式log_a b = (log_c b)/(log_c a)让你可以使用计算器习惯的底数。
4. Trigonometry and Circular Functions | 三角学与圆函数
The three primary ratios are sine (sin = opposite/hypotenuse), cosine (cos = adjacent/hypotenuse), and tangent (tan = opposite/adjacent). Remember SOH CAH TOA. For all angles, we use the unit circle: radians are essential – π rad = 180°, so to convert degrees to radians multiply by π/180. The CAST diagram tells you which ratios are positive in each quadrant (All, Sin, Tan, Cos). Trigonometric identities like sin²θ + cos²θ ≡ 1 and tanθ ≡ sinθ/cosθ are used to simplify expressions and solve equations. The graphs of sin, cos and tan are periodic: sin and cos have period 2π, tan has period π. Amplitude is the height from the centre line, and the phase shift moves the graph horizontally.
三个基本比是正弦（sine, sin = 对边/斜边）、余弦（cosine, cos = 邻边/斜边）和正切（tangent, tan = 对边/邻边）。记住SOH CAH TOA。对于任意角，我们使用单位圆：弧度制（radians）至关重要——π rad = 180°，因此角度转弧度需乘以π/180。 CAST图告诉你每个象限中哪些比为正（全正、Sin、Tan、Cos）。 三角恒等式（Trigonometric identities）如sin²θ + cos²θ ≡ 1和tanθ ≡ sinθ/cosθ，用于化简表达式和求解方程。 sin、cos、tan的图像都是周期函数：sin和cos的周期为2π，tan的周期为π。振幅是从中线起的高度，相位移动（phase shift）将图像水平平移。
5. Vectors in Two Dimensions | 平面向量
A vector has both magnitude and direction, represented as a column vector (x y) or using i, j notation: xi + yj. The magnitude |v| = √x² + y² gives the length. A unit vector has magnitude 1; to find it, divide the vector by its magnitude. Position vector starts at the origin and ends at a point. Displacement vector from A to B is B − A (as vectors). Scalar multiplication stretches or shrinks a vector. Vector addition follows the triangle law. The dot product (scalar product) v·w = |v||w| cos θ = x₁x₂ + y₁y₂. If v·w = 0, the vectors are perpendicular. The angle between vectors can be found using cos θ = (v·w)/(|v||w|).
向量（Vector）既有大小又有方向，可用列向量 (x y) 或i, j符号表示：xi + yj。 模（magnitude）|v| = √x² + y² 给出长度。 单位向量（unit vector）的模为1；求法是将向量除以它的模。 位置向量（position vector）起点在原点，终点在某一点。从A到B的位移向量（displacement vector）是B − A（作为向量运算）。 数乘（scalar multiplication）将向量拉长或缩短。 向量加法（vector addition）遵循三角形法则。 点积（dot product，也称标量积）v·w = |v||w| cos θ = x₁x₂ + y₁y₂。若v·w = 0，则两向量垂直（perpendicular）。两向量之间的夹角可通过cos θ = (v·w)/(|v||w|)求得。
6. Calculus: Differentiation | 微积分：微分
Differentiation finds the gradient of a curve at a point, also called the derivative dy/dx or f'(x). The power rule: if y = xⁿ then dy/dx = nxⁿ⁻¹. The derivative of a constant is 0. For tangent and normal: the tangent's gradient is dy/dx; the normal's gradient is the negative reciprocal (−1/(dy/dx)). Stationary points occur where dy/dx = 0: solve to find them, then use the second derivative d²y/dx² to classify – if d²y/dx² > 0 it's a minimum, if < 0 a maximum, if = 0 could be a point of inflection. Rates of change use connected rates: e.g., dV/dt = (dV/dr)(dr/dt). The chain rule: if y is a function of u and u is a function of x, dy/dx = (dy/du)(du/dx). The product rule: d(uv)/dx = u'v + uv'. The quotient rule: d(u/v)/dx = (u'v − uv')/v².
微分（Differentiation）求曲线在某点的斜率，也称为导数（derivative）dy/dx或f'(x)。幂法则：若y = xⁿ，则dy/dx = nxⁿ⁻¹。常数的导数为0。关于切线（tangent）和法线（normal）：切线的斜率是dy/dx；法线的斜率是其负倒数（−1/(dy/dx)）。 驻点（Stationary points）出现在dy/dx = 0处：先求解得出位置，再用二阶导数（second derivative）d²y/dx²进行分类——若d²y/dx² > 0则为极小值，若< 0则为极大值，若= 0则可能是拐点。 变化率（Rates of change）使用相关变化率：例如dV/dt = (dV/dr)(dr/dt)。 链式法则（chain rule）：若y是u的函数，u是x的函数，则dy/dx = (dy/du)(du/dx)。 积法则（product rule）：d(uv)/dx = u'v + uv'。 商法则（quotient rule）：d(u/v)/dx = (u'v − uv')/v²。
7. Calculus: Integration | 微积分：积分
Integration is the reverse of differentiation; the result is an antiderivative rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, provided n ≠ −1. For n = −1, ∫ x⁻¹ dx = ln |x| + C. Definite integrals evaluate the area under a curve between limits a and b: ∫_ab f(x) dx = F(b) − F(a). The constant C cancels out. Area between two curves is found by integrating the difference of the functions. Kinematics applications: velocity is the integral of acceleration, displacement is the integral of velocity; initial conditions determine constants.
积分（Integration）是微分的逆运算；结果是反导数（antiderivative）或不定期积分，并包含积分常数+C。积分的幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，前提是n ≠ −1。对于n = −1，∫ x⁻¹ dx = ln |x| + C。 定积分（Definite integrals）计算的是曲线在a和b之间的面积：∫_ab f(x) dx = F(b) − F(a)。常数C会抵消。 两曲线之间的面积通过对两个函数的差进行积分求得。 运动学应用（Kinematics applications）：速度是加速度的积分，位移是速度的积分；初始条件决定了常数值。
8. Sequences, Series and Binomial Expansion | 数列、级数与二项式展开
An arithmetic progression (AP) has a common difference d; the nth term is a + (n−1)d, and the sum of the first n terms is Sn = n/2 (2a + (n−1)d) or n/2 (first + last). A geometric progression (GP) has a common ratio r; the nth term is arⁿ⁻¹. The sum of the first n terms is Sn = a(1 − rⁿ)/(1 − r) for r ≠ 1. An infinite geometric series converges to a sum S∞ = a/(1 − r) provided |r| < 1; otherwise it diverges. The binomial expansion for (a + b)ⁿ uses Pascal’s triangle or the binomial coefficient nCr = n!/(r!(n−r)!). For (1 + x)ⁿ where n is a positive integer, the expansion is 1 + nx + n(n−1)/2! ² + ... When n is not a positive integer, the expansion is an infinite series and requires |x| < 1 for validity. Key term: factorial n! = n × (n−1) × ... × 1.
等差数列（arithmetic progression, AP）具有公差d；第n项为a + (n−1)d，前n项和为Sn = n/2 (2a + (n−1)d) 或 n/2 (首项+末项)。 等比数列（geometric progression, GP）具有公比r；第n项为arⁿ⁻¹。前n项和（r ≠ 1）为Sn = a(1 − rⁿ)/(1 − r)。 无穷等比级数（infinite geometric series）在|r| < 1时收敛，和为S∞ = a/(1 − r)；否则发散。 二项式展开（binomial expansion）对于(a + b)ⁿ，可使用帕斯卡三角形或二项式系数nCr = n!/(r!(n−r)!)。对于(1 + x)ⁿ，当n为正整数时，展开式为1 + nx + n(n−1)/2! x² + ... 当n不是正整数时，展开式为无穷级数，并要求|x| < 1才有效。关键术语：阶乘（factorial）n! = n × (n−1) × ... × 1。
9. Coordinate Geometry and Straight Lines | 坐标几何与直线
The gradient of a line joining (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). The equation of a line c (slope-intercept), y − y₁ = m(x − x₁) (point-slope), or ax + by + c = 0 (general form). Parallel lines have equal gradients, perpendicular lines have gradients whose product is −1 (m₁m₂ = −1). The midpoint of two points is ((x₁+x₂)/2, (y₁+y₂)/2). The distance between points is √((x₂−x₁)² + (y₂−y₁)²). For circles r. To find where a line and a circle intersect, substitute the line equation into the circle’s and solve the resulting quadratic.
连接(x₁, y₁)和(x₂, y₂)的直线斜率（gradient）为m = (y₂ − y₁)/(x₂ − x₁)。 直线方程（equation of a line）可用多种形式表示：y = mx + c（斜截式），y − y₁ = m(x − x₁)（点斜式），或ax + by + c = 0（一般式）。 平行线（parallel lines）斜率相等，垂直线（perpendicular lines）斜率的乘积为−1（m₁m₂ = −1）。两点的中点（midpoint）为((x₁+x₂)/2, (y₁+y₂)/2)。两点间的距离（distance）为√((x₂−x₁)² + (y₂−y₁)²)。对于圆（circle），标准方程为(x − h)² + (y − k)² = r²，圆心为(h,k)，半径为r。要求直线与圆的交点，将直线方程代入圆的方程并解所得的二次方程。
10. Polynomials and Factors | 多项式与因式
A polynomial is an expression like an xⁿ + ... + a₁x + a₀. The degree is the highest power of x. The factor theorem states: (x − p) is a factor of f(x) if and only if f(p) = 0. The remainder theorem: when f(x) is divided by (x − p), the remainder is f(p). These two combine to help factorise cubics and higher-degree polynomials. To divide polynomials, use long division or synthetic division. Rational root theorem er Maths, you often need to fully factorise a cubic given one factor, then solve f(x) = 0.
多项式（polynomial）是形如an xⁿ + ... + a₁x + a₀的表达式。 次数（degree）是x的最高次幂。 因式定理（factor theorem）指出：若且唯若f(p) = 0，则(x − p)是f(x)的一个因式。 余式定理（remainder theorem）：当f(x)除以(x − p)时，余式为f(p)。这两者结合有助于对三次及更高次多项式进行因式分解。进行多项式除法（divide polynomials）时可使用长除法或综合除法。 有理根定理（rational root theorem）提示：任意有理根p/q必须满足p整除常数项，q整除首项系数。在CIE进阶数学中，你经常需要根据已知的一个因式对三次多项式进行完全分解，然后解方程f(x) = 0。
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